1. Field of the Invention
The present invention relates to a method, a system, and a computer program product for estimating stock option prices using a combination of risk free and long-term equity-based rates of return, historical volatility of the underlying security, and a conditional probability method of data analysis in the estimation model.
2. Background of the Invention
Stock options give the owner a right to buy or sell a specific number of shares of stock during a time and at a specified price. An option to buy is referred to as a “call option,” and an option to sell is referred to as a “put option.” Stock options always have a start date and an expiration date. If they are not exercised before the expiration date, they are lost. Two styles of options are the European style and the American style. A European style option permits exercise of the option only on the expiration date. In contrast, an American style option permits exercise of the option at any time during its life up to the expiration date. Stock options are publicly traded, and are also commonly granted to employees of the issuing company as part of a compensation package. Generally, stock options which are granted to employees are call options which confers the right to buy stock later in time. The grant or strike price is often the market price of the stock at the time the options are granted.
A simple example regarding employee stock options will illustrate the concept. Assume that a company grants its employee options to buy 1000 shares of stock at $2 a share. The employee can exercise the options starting after a waiting period which is 3 years from the grant date. The expiration date is 4 years from the grant date. On the 3 year date, assume that the stock is at $4 per share. In this case, the employee has three options. The first choice is to convert the options to cash by first purchasing the 1000 shares at $2 per share, and then selling all 1000 shares at $4 per share. The net gain is $2 per share or $2,000. Another choice is to sell some of the shares at $4 per share, and hold the rest to sell later. A third choice is to hold all 1000 shares. If on the 3 year date of the option period, the stock price has fallen below the grant price, the employee would likely let the option expire. For example, if the stock price is at $1, it makes no sense to spend $2 to buy a share that could be bought on the open market for $1.
Accurate option price estimation would help the owner or an options trader to evaluate the risks and benefits of holding the options. Stock option valuation is determined by way of mathematical models. A well known and generally accepted model is the Black-Scholes stock option pricing model published by Fischer Black and Myron Scholes in 1973 in the Journal of Political Economy. Segments of the academic community have been studying the Black-Scholes model, and have recognized some shortcomings of its results, in particular that Black-Scholes results have recently tended to overstate options values for options traded on the open market.
The accuracy of any mathematical model can be confirmed by comparing the predicted outcome using reasonable assumptions with actual observed results. For example, using the data for publicly traded options which are published in financial sections, one can apply a forecast formula and compare the results with the actual published prices. Detailed examples of empirical data comparison using both the Black-Scholes formula and the inventive method are provided in the detailed description.
A relatively recent development brought this generally academic issue to the attention of the financial and business communities. A new accounting rule issued in December of 2004 by the Financial Accounting Standards Board (FASB), FAS 123(R) Share-Based Payment, requires that companies issuing employee stock options must expense the options. FAS 123(R) becomes effective in 2006. This means that all companies issuing employee stock options will be assessed a compensation expense and therefore must assign a value to the options at the time they are issued. With the advent of FAS 123(R), the financial and business communities are now more interested than ever in stock option valuation, and have also come to recognize the deficiencies of a model that overstates options values. Even with its deficiencies, Black-Scholes is so well known, that FAS 123(R) mentions “Black-Scholes” by name as one of the very few techniques of determining a stock option value that meets the specified requirements of FAS 123(R).
The problem with option valuation has been reported repeatedly in the financial press. A sampling of articles and opinions pieces published from 2002-2004 in the Wall Street Journal on the subject all carry the same theme: that Black-Scholes tends to overstate values. “The Options-Value Brain Teaser,” by Jonathan Weil and Theo Francis, published Aug. 6, 2002, Page C1. “Fixing the Numbers Problems,” by Jonathan Weil, published Jan. 13, 2003, Page C1, which states that Black-Scholes makes highly volatile stocks' options appear unduly valuable. “Coke Developed a New Way to Value Options, But Company Will Return to Its Classic Formula,” by Jonathan Weil and Betsy McKay, published Mar. 7, 2003, Page C3. “‘Kind of Right’ Isn't Good Enough,” by Craig R. Barrett, at the time the CEO of Intel, became Chairman of the Board in 2005, which discusses the shortcomings of the Black-Scholes formula and the impact on Intel. “The Stock-Option Showdown,” by Jonathan Weil and Jeanne Cummins, published Mar. 9, 2004, Page C1, which describes the new accounting rules and the opposition of the tech industry. “Shock! The Numbers Are Merely Estimates,” by Jonathan Weil, published Mar. 9, 2004, Page C1, in which the Chief Executive of RSA Security states that the existing models produce inaccurate and misleading information.
Accurately estimating the prices of stock options is obviously useful for an options trader, or a company which issues stock options as part of a compensation program. Anyone holding employee stock options in their investment portfolio would be interested in more accurate valuations to more accurately assess the value of their portfolios. This is important for all financial planning, but particularly for retirement planning. For a company issuing employee stock options, accurate valuation has become even more important with the advent of FAS 123(R), since stock option values must be disclosed in corporate financial statements.
Another reason that more accurate valuations are important is that stock option pricing is a factor that may be used to evaluate the anticipated performance of the underlying stock. In other words, stock option prices contain implicit information regarding the strength of the company as whole. Financial professionals doing their research into companies will be interested in more accurate estimates of stock option pricing.
At the time of its publication in 1973, the Black-Scholes option pricing formula provided a breakthrough theoretical framework for pricing options and other derivative instruments, and launched the field of financial engineering. The Black-Scholes formula prices European call or put options on a stock that is assumed to pay no dividends or make other distributions. The formula assumes that the underlying stock price follows a geometric Brownian motion, and is developed using a partial differential equation for valuing claims contingent on the underlying stock price. In the original Black-Scholes formula, values for a call price c or put price p are:c=SN(d1)−Ke−rtN(d2)p=Ke−rtN(−d2)−SN(−d1)
where:
      d    1    =                    ln        ⁡                  (                      S            /            K                    )                    +                        (                      r            +                                          σ                2                            /              2                                )                ⁢        t                    σ      ⁢              t            d2=d1−σ√{square root over (t)}
And the variables denote the following:
S=the price of the underlying stock;
K=the strike price;
r=the current continuously compounded risk free interest rate;
t=the time in years until the expiration of the option;
σ=the estimated volatility for the underlying stock; and
N=the standard normal cumulative distribution function.
Since the original Black-Scholes formula was published, the academic financial community has recognized that the formula has become inaccurate, even in markets where one could expect it to be most accurate. Numerous experts have worked to try to pinpoint the issues in Black-Scholes which may be responsible for the problematic results. For the most part, the experts have focused on σ, the volatility parameter. Volatility is quantified as a measure of the degree to which the price of the underlying stock tends to fluctuate over time, and is expressed in decimal form, for example, an annualized volatility of 10% is expressed as 0.10. Volatility estimates for the future have generally been based on historic volatility. One approach to correcting the Black-Scholes accuracy problems resulted in an extension of the Black-Scholes model using a parameter labeled “stochastic volatility” which is a modified way of quantifying and calculating volatility. Despite its recognized shortcomings, for lack of a better model, Black-Scholes or some modification of it, continues to be used widely.
One of the underlying deficiencies of conventional stock option pricing models is the use of historical investment return data as if the data points were independent events for the purpose of statistical analysis. However, the historical performance of the stock market may more correctly be viewed as a single event, and data more correctly viewed as periodic observations in that single event. More detailed analysis of this particular shortcoming is addressed in an article authored by the inventor, “Correcting the Overstatement in Investment Forecasts,” published in the Journal of Financial and Economic Practice, Vol. 3, No. 1, Fall 2004, the entire contents of which is hereby incorporated by reference. Further illumination on the use of conditional probabilities for investment forecast modeling is described in another article authored by the inventor, “The Tendency of the Arithmetic Mean to Overstate Expected Returns,” published in the Journal of Financial and Economic Practice, Vol. 6, No. 1, Fall 2005, the entire contents of which is also hereby incorporated by reference.
It is generally accepted that adjustments to the conventional models which address the volatility parameter do not seem to produce consistently better results.
Another problematic feature of the Black-Scholes model was setting the discount rate at the risk-free rate of return. The original presentation included in the Black-Scholes 1973 paper had an unstated assumption that there is one and only one investment return parameter. The presentation thus precluded the possibility of a stock option pricing model that reflects the dynamic interaction of both risk-free and long term equity based rates of return.
There has been a need for a method for stock option valuation that more accurately matches listed option prices when compared with empirical data.